Study Guide Block 1: Vector Arithmetic Unit 6: Equations of Lines and Planes 1.
Lecture 1.060 Study Guide
Block 1: V e c t o r A r i t h m e t i c
U n i t 6: E q u a t i o n s o f L i n e s and P l a n e s
Study Guide
Block 1: Vector A r i t h m e t i c
U n i t 6: Equations o f Lines and P l a n e s
2. Read Thomas, s e c t i o n 12.8
3. E x e r c i s e s :
1.6.1(L)
Find t h e e q u a t i o n of t h e p l a n e determined by t h e p o i n t s A(1,2,3),
3 (3,3,5)
, and
C (4,8,1).
(see ~ x e r c i s e1.5.2)
Show t h a t y = 2x i s t h e e q u a t i o n o f t h e p l a n e which p a s s e s through
( 0 , 0 , 0 ) a n d h a s 2;
-
$
a s i t s normal.
'
,
a. What i s t h e e q u a t i o n o f t h e l i n e which i s p a r a l l e l t o 3 1
+
45
+
2);
and p a s s e s through ( - 2 , 5 , - l ) ?
b. A t what p o i n t does t h i s l i n e i n t e r s e c t t h e xy-plane?
1.6.4 Find t h e d i r e c t i o n a l c o s i n e s f o r t h e l i n e 1.6.5 33 + 5% and
i n t e r s e c t t h e plane given i n
A t what p o i n t does t h e l i n e which is p a r a l l e l t o 2x
which p a s s e s through (-1,3,-2)
+
E x e r c i s e 1.6-13
1.6.6tL)
a . Find t h e d i s t a n c e from ( 7 , 8 , 9 ) t o t h e p l a n e 2x
+
3y
+
62 = 8.
b. A t what p o i n t does t h e l i n e through ( 7 , 8 , 9 ) p e r p e n d i c u l a r t o t h e
p l a n e i n a. i n t e r s e c t t h i s p l a n e ?
c . Find t h e d i s t a n c e between t h e p l a n e s 2x
2x
+
3y
+
+
3y
+
6 2 = 22 and
9
62 = 8.
Study Guide
Block 1: v e c t o r A r i t h m e t i c
Unit 6: Equations of Lines and P l a n e s
The p o i n t s A(1,1,4) and B(3,4,5) a r e on t h e l i n e L, w h i l e t h e
points C(3,4,-l),
D ( 4 , 6 , 2 ) , and E ( 8 , 9 , 7 ) a r e i n t h e p l a n e M.
At
what p o i n t does t h e l i n e L i n t e r s e c t t h e p l a n e M?
a.
Find t h e a n g l e between t h e p l a n e s whose C a r t e s i a n e q u a t i o n s a r e
3x
b.
+
2y
+
62 = 8 and 2x
+
2y
-
z = 5.
,
.
Find t h e C a r t e s i a n e q u a t i o n o f t h e l i n e which i s t h e i n t e r s e c t i o n
of t h e two p l a n e s given i n p a r t a .
Study Guide lock 1: Vector A r i t h m e t i c Quiz
1. By computing (-1)(1
-
1) i n two d i f f e r e n t ways, use t h e r u l e s and
theorems of a r i t h m e t i c t o prove t h a t (-I)(-1)= 1.
2. Find a u n i t v e c t o r which i s normal t o t h e curve y = x3
+
x a t the
point (1,2).
3. L e t A(1,3,5)
, B (3,4,7)
and C ( - 1 , O ,-1) be p o i n t s i n space.
Find
a u n i t v e c t o r such t h a t i t o r i g i n a t e s a t A; l i e s i n t h e p l a n e
determined by A , B, and C; and b i s e c t s 4BAC.
4. '
..
Let P denote t h e plane determined by t h e p o i n t s A ( 1 , 2 , 3 ) , B ( 3 , 3 , 5 ) ,
and ( 4 , 4 , 9 ) .
(a)
Determine t h e measure of 3BAC.
Ib)
Find t h e equation of t h e plane P. (c) What i s t h e a r e a of AABC?
.-
(d) A t what p o i n t do t h e medians of AABC i n t e r s e c t ?
( e ) What i s t h e equation of t h e l i n e which passes through C and
is p a r a l l e l t o AB?
. .
5 . Find t h e d i s t a n c e of t h e p i n t p0(2 , 3 , 4 ) from t h e p l a n e whose
C a r t e s i a n e q u a t i o n i s 4x + 5 y + 2 2 = 6.
A l s o determine t h e
p o i n t a t which t h e l i n e through Po, perpendicular t o t h e plane
4x + 5y + 2 2 = 6 , i n t e r s e c t s this plane.
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Resource: Calculus Revisited: Multivariable Calculus
Prof. Herbert Gross
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